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| Main Authors: | , |
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| Format: | Preprint |
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2026
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| Online Access: | https://arxiv.org/abs/2604.15157 |
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| _version_ | 1866914480273752064 |
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| author | Bazzanella, Alice Sanna, Carlo |
| author_facet | Bazzanella, Alice Sanna, Carlo |
| contents | Let $N$ be a positive integer and let $S_N$ be the set of polynomials with integer coefficients, degree less than $N$, and minimal positive integral over $[0,1]$. D. Bazzanella initiated the study of $S_N$ because of its relation to the distribution of prime numbers. Indeed, it is possible to prove that $\sum_{p^m \leq N} \log p = -\log \int_0^1 P(x) \mathrm{d} x$ for every $P \in S_N$, where the sum runs over prime numbers $p$ and positive integers $m$ such that $p^m \leq N$. For each real number $t$, let $\lfloor t \rfloor$ denote the maximal integer not exceeding $t$. The main result of this paper states that there exist infinitely many polynomials $P \in S_N$ such that $\big(x^3(1 - x)^2\big)^{\lfloor N / 6 \rfloor}$ divides $P(x)$ in $\mathbb{Z}[x]$. This improves upon a similar result of Sanna, who proved the same claim but with the lower-degree polynomial $\big(x(1-x)\big)^{\lfloor N / 3 \rfloor}$ in place of $\big(x^3(1 - x)^2\big)^{\lfloor N / 6 \rfloor}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_15157 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Another factor of integer polynomials with minimal integrals Bazzanella, Alice Sanna, Carlo Number Theory Let $N$ be a positive integer and let $S_N$ be the set of polynomials with integer coefficients, degree less than $N$, and minimal positive integral over $[0,1]$. D. Bazzanella initiated the study of $S_N$ because of its relation to the distribution of prime numbers. Indeed, it is possible to prove that $\sum_{p^m \leq N} \log p = -\log \int_0^1 P(x) \mathrm{d} x$ for every $P \in S_N$, where the sum runs over prime numbers $p$ and positive integers $m$ such that $p^m \leq N$. For each real number $t$, let $\lfloor t \rfloor$ denote the maximal integer not exceeding $t$. The main result of this paper states that there exist infinitely many polynomials $P \in S_N$ such that $\big(x^3(1 - x)^2\big)^{\lfloor N / 6 \rfloor}$ divides $P(x)$ in $\mathbb{Z}[x]$. This improves upon a similar result of Sanna, who proved the same claim but with the lower-degree polynomial $\big(x(1-x)\big)^{\lfloor N / 3 \rfloor}$ in place of $\big(x^3(1 - x)^2\big)^{\lfloor N / 6 \rfloor}$. |
| title | Another factor of integer polynomials with minimal integrals |
| topic | Number Theory |
| url | https://arxiv.org/abs/2604.15157 |